College Algebra, 2013a

6.3 Exponential Equations and Inequalities 449
1. Since16isapowerof2,wecanrewrite2 3x =16 1−x as 2 3x = ( 2 4) 1−x . Using properties of
exponents, we get 2 3x =2 4(1−x) . Using the one-to-one property of exponential functions, we
get 3x =4(1−x) which gives x = 4 7 . To check graphically, we set f(x) =23x and g(x) =16 1−x
and see that they intersect at x = 4 7 ≈ 0.5714.
2. We begin solving 2000 = 1000·3 −0.1t by dividing both sides by 1000 to isolate the exponential
which yields 3 −0.1t = 2. Since it is inconvenient to write 2 as a power of 3, we use the natural
log to get ln ( 3 −0.1t) = ln(2). Using the Power Rule, we get −0.1t ln(3) = ln(2), so we
divide both sides by −0.1 ln(3) to get t = − ln(2) 10 ln(2)
0.1 ln(3)
= −
ln(3)
. On the calculator, we graph
f(x) = 2000 and g(x) = 1000 · 3 −0.1x and find that they intersect at x = − ≈−6.3093.
10 ln(2)
ln(3)
y = f(x) =2 3x and
y = g(x) =16 1−x
y = f(x) = 2000 and
y = g(x) = 1000 · 3 −0.1x
3. We first note that we can rewrite the equation 9·3 x =7 2x as 3 2·3 x =7 2x to obtain 3 x+2 =7 2x .
Since it is not convenient to express both sides as a power of 3 (or 7 for that matter) we use
the natural log: ln ( 3 x+2) =ln ( 7 2x) . The power rule gives (x +2)ln(3) = 2x ln(7). Even
though this equation appears very complicated, keep in mind that ln(3) and ln(7) are just
constants. The equation (x +2)ln(3)=2x ln(7) is actually a linear equation and as such we
gather all of the terms with x on one side, and the constants on the other. We then divide
both sides by the coefficient of x, which we obtain by factoring.
(x +2)ln(3) = 2x ln(7)
x ln(3) + 2 ln(3) = 2x ln(7)
2ln(3) = 2x ln(7) − x ln(3)
2 ln(3) = x(2 ln(7) − ln(3)) Factor.
x =
2 ln(3)
2 ln(7)−ln(3)
Graphing f(x) =9·3 x and g(x) =7 2x on the calculator, we see that these two graphs intersect
at x =
2 ln(3)
2 ln(7)−ln(3) ≈ 0.7866.
4. Our objective in solving 75 = 100 is to first isolate the exponential. To that end, we
1+3e −2t
clear denominators and get 75 ( 1+3e −2t) = 100. From this we get 75 + 225e −2t = 100,
which leads to 225e −2t = 25, and finally, e −2t = 1 9
. Taking the natural log of both sides